The synthesis and properties of the hybrid organic/inorganic germanium perovskite compounds, AGeI3, are reported (A = Cs, organic cation). The systematic study of this reaction system led to the isolation of 6 new hybrid semiconductors. Using CsGeI3 (1) as the prototype compound, we have prepared methylammonium, CH3NH3GeI3 (2), formamidinium, HC(NH2)2GeI3 (3), acetamidinium, CH3C(NH2)2GeI3 (4), guanidinium, C(NH2)3GeI3 (5), trimethylammonium, (CH3)3NHGeI3 (6), and isopropylammonium, (CH3)2C(H)NH3GeI3 (7) analogues. The crystal structures of the compounds are classified based on their dimensionality with 1-4 forming 3D perovskite frameworks and 5-7 1D infinite chains. Compounds 1-7, with the exception of compounds 5 (centrosymmetric) and 7 (nonpolar acentric), crystallize in polar space groups. The 3D compounds have direct band gaps of 1.6 eV (1), 1.9 eV (2), 2.2 eV (3), and 2.5 eV (4), while the 1D compounds have indirect band gaps of 2.7 eV (5), 2.5 eV (6), and 2.8 eV (7). Herein, we report on the second harmonic generation (SHG) properties of the compounds, which display remarkably strong, type I phase-matchable SHG response with high laser-induced damage thresholds (up to 3 GW/cm2). The second-order nonlinear susceptibility, S(2), was determined to be 125.3 ± 10.5 pm/V (1), (161.0 ± 14.5) pm/V (2), 143.0 ± 13.5 pm/V (3), and 57.2 ± 5.5 pm/V (4). First-principles density functional theory electronic structure calculations indicate that the large SHG response is attributed to the high density of states in the valence band due to sp-hybridization of the Ge and I orbitals, a consequence of the lone pair activation.